Especially on the smaller stackexchange sites, the main page's active list is the main place to go to see the new activity. However sometimes, new answers and questions are lost, or flushed through too quickly by much retagging of old questions. I'm all for retagging old questions as it helps keep the site organized, but it would be nice if I could view activity excluding those posts that have no meaningful content change. I know that just changing the tags doesn't count as an edit (say, for the Strunk and White badge) so maybe we can have a way of viewing recently edited posts in addition to recently active posts.
There's currently no way to retag without bumping, and sometimes such retagging is necessary.
Often, though, what you want to accomplish by retagging can be done (and much more easily!) by a ♦ moderator by merging tags, which does not bump the questions as active. If you ever find yourself wanting to retag a bunch of questions the same way, consider asking a mod in chat whether it can be done by merging instead, or propose it as a question here on meta.
When retagging really is necessary, I won't agree merely-retagged posts shouldn't appear as active: sometimes such retaggings need to be checked for accuracy. Perhaps, though, retaggings by a high-rep user should not bump questions?
Note also that in any specific case of mass-retagging, you can (temporarily if you like) set the questions' new tag as "ignored" (on the homepage, on the right), which will grey them out on the active-questions list.
I think older questions that do not have an accepted answer should be bumped with any activity. This gives fresh eyes a chance to review and possibly answer the question.